Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF

Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories (e.g., NF and Church's CUS) which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for “low” sets, and thus, for example, a set of all (low) ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no set of all ordinals is obtained thereby. This “recurrence problem” is discussed.